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Physica Scripta

The spin-1/2 Ising film with a perfect surface

A. Oubelkacem, A. Ainane, M. Saber, P. Häussler

Published Mar 3, 2006• 4 citations
Publisher

Abstract

The critical properties of the magnetic spin-1/2 Ising film are studied using the effective field theory with a probability distribution technique that accounts for the self-spin correlation functions. We discuss an L -layer film of simple cubic symmetry with nearest-neighbour interactions in which the strength of the interaction at the surface is different from the bulk value. We derive the phase diagram, the longitudinal susceptibility and the effective critical exponents in the case of a perfect surface. In such films, the critical temperature can shift to either lower or higher temperature compared to the corresponding bulk value.

DOI
10.1088/0031-8949/73/4/003

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The critical properties of the magnetic spin-1/2 Ising film are studied using the effective field theory with a probability distribution technique that accounts for the self-spin correlation functions. We discuss an L -layer film of simple cubic symmetry with nearest-neighbour interactions in which the strength of the interaction at the surface is different from the bulk value. We derive the phase diagram, the longitudinal susceptibility and the effective critical exponents in the case of a perfect surface. In such films, the critical temperature can shift to either lower or higher temperature compared to the corresponding bulk value.
Within the framework of the effective-field theory, we examine the phase transitions of a transverse spin-12\frac{1}{2}21​ Ising film. The critical temperatures of the film as a function of the interactions, transverse fields, and film thickness are studied. It is found that for the ratio of the surface interactions to the bulk ones R=Js/J{R=J}_{s}/JR=Js​/J less than a critical value Rc,{R}_{c},Rc​, the critical temperature Tc/J{T}_{c}/JTc​/J of the film is smaller than the bulk critical temperature TcB/J{T}_{c}^{B}/JTcB​/J and as the film thickness L is increased further, Tc/J{T}_{c}/JTc​/J increases and approaches asymptotically TcB/J{T}_{c}^{B}/JTcB​/J for large values of L. However, for R>Rc,Tc/JR>{R}_{c}, {T}_{c}/JR>Rc​,Tc​/J is larger both than the bulk TcB/J{T}_{c}^{B}/JTcB​/J and the surface TcS/J{T}_{c}^{S}/JTcS​/J critical temperatures of the corresponding semi-infinite system and as the film thickness L is increased further, Tc/J{T}_{c}/JTc​/J decreases and approaches asymptotically, for large values of L, the surface magnetic transition TcS/J{T}_{c}^{S}/JTcS​/J observed in the corresponding semi-infinite system. We calculate also some magnetic properties of the film such as the layer magnetizations, their averages and their profiles and the longitudinal susceptibility of the film. The film longitudinal susceptibility still diverges at the film critical temperature as does the bulk longitudinal susceptibility, but its magnitude is reduced. Also there is a rounded peak at the bulk critical temperature when R>Rc.R>{R}_{c}.R>Rc​. The bulk related character of the susceptibility is more pronounced and the surface related character is less pronounced when the film thickness is large.
Within the framework of the effective field theory, with a probability distribution technique that accounts for the single-site spin correlations, we examine the critical behavior of a diluted spin-1/2 Ising film. The critical temperature of the system is studied as a function of the thickness of the film, the concentration of magnetic atoms, the strength of the transverse field and the couplings. It is shown that for the ratio of the surface couplings to the bulk ones R = Js/J less than a critical value R,, the critical temperature of the film Tc/J is smaller than the bulk critical temperature Tc^B/J and for R ≥ Rc, Tc/J is larger than the bulk critical temperature Tc^B/J and Tc^S/J of the corresponding semi-infinite system. As the film thickness L is increased further Tc/J increases for R ≤ Rc (decreases for R 2 R,) and approaches asymptotically for large values of L the bulk critical temperature Tc^B/J (the surface critical temperature Tc^S/J) of the corresponding semi-infinite system.
An effective-field theory that accounts for the self-spin correlation function is used to study the effect of surface dilution on the critical behavior of a ferromagnetic Ising film in transverse surface and bulk fields. Surface exchange enhancement (or reduction) is considered. It is shown that the surface and bulk fields affect the critical ratio of surface to bulk interactions, at which the surface orders before the bulk, in opposite ways. Although it is found that for the undiluted film this leads to a critical ratio that is practically insensitive to field for common bulk and surface field strengths, this feature is relaxed to a certain degree with surface dilution. The thickness dependence of the critical temperature, and phase diagrams in the fields, interaction strength ratio, and dilution parameter space, are presented.
We have used Monte Carlo simulations and mean-field analysis to observe the magnetic behavior of Ising thin films with cubic lattice structures as a function of temperature and thickness, especially in the critical region. Magnetization and magnetic susceptibility, including layer variation, are investigated. We find that the magnetic behavior changes from two-dimensional to three-dimensional character with increasing film thickness. Both the crossover of the critical temperature from a two-dimensional to a bulk value and the shift exponent are observed. Nevertheless, with support from a scaling function, the simulations show that the effective critical exponents for films with large enough layer extents only vary a little from their two-dimensional values. This, in particular, provides an indication of two-dimensional universality in the thin films.